Descent Monomials, P-Partitions and Dense Garsia-Haiman Modules
Edward E. Allen
DOI: 10.1023/B:JACO.0000047281.84115.b7
Abstract
A two-variable analogue of the descents monomials is defined and is shown to form a basis for the dense Garsia-Haiman modules. A two-variable generalization of a decomposition of a P-partition is shown to give the algorithm for the expansion into this descent basis. Some examples of dense Garsia-Haiman modules include the coinvariant rings associated with certain complex reflection groups.
Pages: 173–193
Keywords: descent monomials; $P$-partitions; garsia-haiman modules; convariant rings
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References
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2. E.E. Allen, “Some graded representations of the complex reflection groups,” J. Combin. Theory, Ser. A 87 (1999), 287-332.
3. E.E. Allen, “Bitableaux bases for some Garsia-Haiman modules and other related modules,” Elect. J. Combin. 9 (2002), R36, 1-59.
4. A.M. Garsia, “Combinatorial methods in the theory of Cohen-Macaulay rings,” Adv. Math. 38 (1980), 229- 266.
5. A.M. Garsia and M.D. Haiman, “A graded representation model for Macdonald's polynomials,” Proc. Natl. Acad. Sci. USA 90 (April 1993), 3607-3610.
6. D.E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, Addison-Wesley, Reading, MA, 1968.
7. H. Morita and H.-F. Yamada, “Higher Specht polynomials for the complex reflection group G(r, p, n),” Hokkaido Mathematical Journal 27 (1998), 505-515.
8. V. Riener, “Quotients of Coxeter Complexes and P-partitions,” Mem. Amer. Math Soc. 95 (460), 1992.
9. B.E. Sagan, The Symmetric Group, Wadsworth & Brooks/Cole, Monterey, 1991.
10. R.P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Monterey, 1986.
11. J.R. Stembridge, “On the eigenvalues of representations of reflection groups and wreath products,” Pacific J. Math. 140 (1989), 353-396.